Self-confirming Equilibrium

A self-con rming equilibrium is the answer to the following question: what are the possible limiting outcomes of purposeful interactions among a collection of adaptive agents, each of whom averages past data to approximate moments of the conditional probability distributions of interest? If outcomes converge, a Law of Large Numbers implies that agents' beliefs about conditional moments become correct on events that are observed su ciently often. Beliefs are not necessarily correct about events that are infrequently observed. Where beliefs are correct, a self-con rming equilibrium is like a rational expectations equilibrium. But there can be interesting gaps between self-con rming and rational expectations equilibria where beliefs of some important decision makers are incorrect. Self-con rming equilibria interest macroeconomists because they connect to an in uential 1970s argument made by Christopher Sims that advocated rational expectations as a sensible equilibrium concept. This argument defended rational expectations equilibria against the criticism that they require that agents `know too much' by showing that we do not have to assume that agents start out `knowing the model'. If agents simply average past data, perhaps conditioning by grouping observations, their forecasts will eventually become unimprovable. Research on adaptive learning has shown that that the glass is `half-full' and `half-empty' for this clever 1970s argument. On the one hand, the argument is correct when applied to competitive or in nitesimal agents: by using naive adaptive learning schemes (various versions of recursive least squares), agents can learn every conditional distribution that they require to play best responses within an equilibrium. On the other hand, large agents (e.g., governments in macro models) who can in uence the market outcome cannot expect to learn everything that they need to know to make good decisions: in a self-con rming equilibrium, large agents may base their decisions on conjectures about o -equilibriumpath behaviors which turn out to be incorrect. Thus, a rational expectations equilibrium is a self-con rming equilibrium, but not vice versa. While agents' beliefs can be incorrect o the equilibrium path, the self-con rming equilibrium path still restricts them in interesting ways. For macroeconomic applications, the government's model must be such that its o -equilibrium path beliefs rationalize the decisions (its Ramsey policy or Phelps policy, in the language of [16]) that are revealed